Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
762 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×D32 |
| commutator, socle, cosocle | C23×C32 | C23 |
| Fitting | C22×C32×C4 | C22 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #392 |
| LCS | G ▷ C23×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C4⋊(C3⋊D3) | C2 | semidirect |
| 2. C22×#626 | C2 | semidirect |
| 3. C2×C3×(C2×D3)⋊C4 | C2 | semidirect |
| 4. C2×#3722 | C2 | semidirect |
| 5. C3×C3⋊#95 | C2 | semidirect |
| 28. C2×(C2×D3)⋊C4 | D3 | semidirect |
| 30. C3⋊#95 | D3 | semidirect |
| 38. C22×C3⋊D3 | D4 | semidirect |
| 43. C22×C3×D3 | D4 | semidirect |
| 45. C2×C32×C4 | D4 | semidirect |
| 51. C2×C3×Dic3 | D4 | semidirect |
| 140. C22×D3 | D12 | semidirect |
| 150. C2×C3×C4 | D12 | semidirect |
| 151. C2×C3×C4 | #67 | semidirect |
| 157. C2×Dic3 | D12 | semidirect |
| 234. C32 | #392 | semidirect |
| 256. C2×C4 | #626 | semidirect |
| 298. C3 | #4671 | semidirect |
| 299. C3 | #4960 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊(C3⋊D3) | 2 |
| C22×#626 | 2 |
| C2×C3×(C2×D3)⋊C4 | 2 |
| C2×#3722 | 2 |
| C3×C3⋊#95 | 2 |
| #4960 | 3 |
| #4671 | 3 |